1. Pythagoras

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    Square roots

    Solving equations of the form

    The theorem of Pythagoras

    The converse of the Pythagorean theorem

    Pythagorean triples

    Problem solving using Pythagoras

    Three dimensional problems (Extension)

    x k2

    Contents:

    1Pythagoras

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    Y:\HAESE\SA_09-6ed\SA09-6_01\011SA09-6_01.CDR Wednesday, 18 October 2006 2:11:40 PM PETERDELL

  • OPENING PROBLEM

    HISTORICAL NOTE

    12 PYTHAGORAS (Chapter 1)

    Over 3000 years ago the Egyptians were familiar withthe fact that a triangle with sides in the ratio 3 : 4 : 5 wasa right angled triangle. They used a loop of rope with

    12 knots equally spaced along it to make right angledcorners in their building construction.

    Around 500 BC, the Greek mathematician Pythagorasfound a rule which connects the lengths of the sides

    of any right angled triangle. It is thought that he dis-

    covered the rule while studying tessellations of tiles on

    bathroom floors.

    Such patterns, like the one illustrated were common on

    the walls and floors of bathrooms in ancient Greece.

    The discovery of Pythagoras Theorem led to the discovery of a different type of number

    which does not have a terminating or recurring decimal value and yet has a distinct place

    on the number line. These sorts of numbers likep

    13 are called surds and are irrationalnumbers.

    Consider the following questions:

    1 If the fire truck parks so that the base of the ladder

    is 7 m from the building, will it reach the third floorwindow which is 26 m above the ground?

    2 The second floor window is 22 m above the ground.By how much should the length of the ladder be re-

    duced so that it just reaches the second floor win-

    dow?

    3

    The study of Pythagoras Theorem and surds will enable you to solve questions such as

    these.

    corner

    line of one side of building

    make rope taut

    take hold of knots at arrows

    26 m

    22 m

    2 m

    25 m

    7 m

    A building is on fire and the fire department are called. They need to

    evacuate people from the third floor and second floor windows with the

    extension ladder attached to the back of the truck. The base of the ladder is

    m above the ground. The fully extended ladder is m in length.2 25

    For many centuries people have found it

    necessary to be able to form right angled

    corners. Whether this was for constructing

    buildings or dividing land into rectangular

    fields, the problem was overcome by rela-

    tively simple means.

    If the ladder gets stuck in the fully extended position, how much further should the

    fire truck move from the building to just reach the second floor window?away

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    Y:\HAESE\SA_09-6ed\SA09-6_01\012SA09-6_01.CDR Thursday, 21 September 2006 4:17:46 PM PETERDELL

  • PYTHAGORAS (Chapter 1) 13

    Recall that 5 5 can be written as 52 (five squared) and 52 = 25.Thus, we say the square root of 25 is 5 and write

    p25 = 5.

    Note:p

    reads: the square root of .

    Finding the square root of a number is the opposite of squaring a number.

    To find the square root of 9, i.e.,p9, we need to find a positive number which when

    multiplied by itself equals 9.

    As 3 3 = 9, the number is 3, i.e., p9 = 3:

    Notice that: The symbol p is called the square root sign. pa has meaning for a > 0, and is meaningless if a < 0. pa > 0 (i.e., is positive or zero).

    1 Evaluate, giving a reason:

    ap16 b

    p36 c

    p64 d

    p144

    ep81 f

    p121 g

    p4 h

    p1

    ip0 j

    q1

    9k

    q1

    49l

    q25

    64

    Finding the square root of numbers which are square numbers such as 1, 4, 9, 25, 36, 49etc. is relatively easy. What about something like

    p6?

    What number multiplied by itself gives 6? If we use a calculator ( 6 ) we get2:449 489 743.

    Here we have a non-recurring decimal which is only an approximation ofp6.

    p6 is an

    example of an irrational number, as it cannot be written as a fraction of any two integers.

    In this chapter we will deal with irrational numbers of square root form called surds.

    IRRATIONAL NUMBERS

    EXERCISE 1A

    SQUARE ROOTSA

    Evaluate, giving a reason: ap49 b

    q1

    16

    a As 7 7 = 49 b As 14 1

    4= 1

    16

    thenp49 = 7 then

    q1

    16= 1

    4

    Self TutorExample 1

    =

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    Y:\HAESE\SA_09-6ed\SA09-6_01\013SA09-6_01.CDR Wednesday, 23 August 2006 10:54:18 AM DAVID3

  • 14 PYTHAGORAS (Chapter 1)

    Between which two consecutive integers doesp11 lie?

    Sincep9 = 3 and

    p16 = 4 and 9 < 11 < 16

    thenp9 0.

    Self TutorExample 12

    EXERCISE 1E

    PYTHAGOREAN TRIPLESE

    5

    4

    3

    Let be

    the largestnumber!

    c

    a

    bc

    is not a Pythagorean triple as these numbers are not all integers.

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    Y:\HAESE\SA_09-6ed\SA09-6_01\023SA09-6_01.CDR Friday, 15 September 2006 11:01:40 AM PETERDELL

  • INVESTIGATION 4 PYTHAGOREAN TRIPLES SPREADSHEET

    24 PYTHAGORAS (Chapter 1)

    The result in question 3 enables us to write down many Pythagorean triples using multiples.

    It also enables us to find the lengths of sides for triangles where the sides may not be integers.

    4 Use multiples to write down the value

    of the unknown for the following triangles.

    Given:

    a b c d

    5 Use multiples to write down the value of

    the unknown for the following triangles.

    Given:

    a b c d

    The best known Pythagorean triple is f3, 4, 5g.Other triples are f5, 12, 13g, f7, 24, 25g and f8, 15, 17g.

    There are several formulae that will generate Pythagorean triples.

    For example, 3n, 4n, 5n where n is a positive integer. A spreadsheet will quicklygenerate sets of triples using these formulae.

    SPREADSHEET

    5

    4

    3

    x

    8

    6

    25

    x

    15

    30

    x

    18

    x

    2

    1.5

    Use multiples to write down the

    value of the unknown for the

    following triangles.

    Given:

    a b

    a 5 : 12 : 13 = 10 : x : 26

    ) x = 12 2i.e., x = 24

    b 5 : 12 : 13 = 50 : 120 : y

    ) y = 13 10i.e., y = 130

    Self TutorExample 13

    12

    135

    x

    2610

    y 50

    120

    2 10

    2 10

    1

    p2

    2

    x

    3x x

    p210

    x

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    Y:\HAESE\SA_09-6ed\SA09-6_01\024SA09-6_01.CDR Thursday, 24 August 2006 2:00:48 PM DAVID3

  • PYTHAGORAS (Chapter 1) 25

    Right angled triangles occur in many realistic problem solving situations. When this happens,

    the relationships involved in Pythagoras Theorem can be used as an aid in solving the

    problem.

    The problem solving approach usually involves the following steps:

    Step 1: Draw a neat, clear diagram of the situation.

    Step 2: Mark known lengths and right angles on the diagram.

    Step 3: Use a symbol, such as x, to represent the unknown length.

    Step 4: Write down Pythagoras Theorem for the given information.

    Step 5: Solve the equation.

    Step 6: Write your answer in sentence form (where necessary).

    PROBLEM SOLVING USING PYTHAGORASF

    filldown

    1 Open a new spreadsheet and

    enter the following:

    2 Highlight the formulae in B2, C2 and D2, andfill these down to row 3. Now highlight all theformulae in row 3 and fill down to row 11.

    This should generate 10 sets of Pythagoreantriples that are multiples of 3, 4 and 5.

    3 Change the headings in B1 to 5n, C1 to 12n and D1 to 13n.

    Now change the formulae in B2 to =A2*5 in C2 to =A2*12 in D2 to =A2*13:

    Refill these to row 11. This should generate 10 more Pythagorean triples that are mul-tiples of 5, 12 and 13.

    4 Manipulate your spreadsheet to find other sets of Pythagorean triples based on:

    a f7, 24, 25g b f8, 15, 17gExtension:

    5 The formulae 2n+1, 2n2+2n, 2n2+2n+1 will generate Pythagorean triples.Devise formulae to enter in row 2 so your spreadsheet calculates the triples.

    6 How could you check that the triples generated above are Pythagorean triples using

    your spreadsheet? Devise suitable formulae to do this.

    7 Use a graphics calculator to generate sets of Pythagorean triples in a TABLE OR

    LIST.

    What to do:

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